gp: [N,k,chi] = [315,2,Mod(46,315)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
sage: from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(315, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 2, names="a")
magma: //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("315.46");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: traces = [4,2]
f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
gp: f = lf[1] \\ Warning: the index may be different
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the q q q -expansion are expressed in terms of a basis 1 , β 1 , β 2 , β 3 1,\beta_1,\beta_2,\beta_3 1 , β 1 , β 2 , β 3 for the coefficient ring described below.
We also show the integral q q q -expansion of the trace form .
Basis of coefficient ring in terms of a root ν \nu ν of
x 4 + 2 x 2 + 4 x^{4} + 2x^{2} + 4 x 4 + 2 x 2 + 4
x^4 + 2*x^2 + 4
:
β 1 \beta_{1} β 1 = = =
ν \nu ν
v
β 2 \beta_{2} β 2 = = =
( ν 2 ) / 2 ( \nu^{2} ) / 2 ( ν 2 ) / 2
(v^2) / 2
β 3 \beta_{3} β 3 = = =
( ν 3 ) / 2 ( \nu^{3} ) / 2 ( ν 3 ) / 2
(v^3) / 2
ν \nu ν = = =
β 1 \beta_1 β 1
b1
ν 2 \nu^{2} ν 2 = = =
2 β 2 2\beta_{2} 2 β 2
2*b2
ν 3 \nu^{3} ν 3 = = =
2 β 3 2\beta_{3} 2 β 3
2*b3
Character values
We give the values of χ \chi χ on generators for ( Z / 315 Z ) × \left(\mathbb{Z}/315\mathbb{Z}\right)^\times ( Z / 3 1 5 Z ) × .
n n n
127 127 1 2 7
136 136 1 3 6
281 281 2 8 1
χ ( n ) \chi(n) χ ( n )
1 1 1
− 1 − β 2 -1 - \beta_{2} − 1 − β 2
1 1 1
For each embedding ι m \iota_m ι m of the coefficient field, the values ι m ( a n ) \iota_m(a_n) ι m ( a n ) are shown below.
For more information on an embedded modular form you can click on its label.
gp: mfembed(f)
Refresh table
This newform subspace can be constructed as the kernel of the linear operator
T 2 4 − 2 T 2 3 + 5 T 2 2 + 2 T 2 + 1 T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1 T 2 4 − 2 T 2 3 + 5 T 2 2 + 2 T 2 + 1
T2^4 - 2*T2^3 + 5*T2^2 + 2*T2 + 1
acting on S 2 n e w ( 315 , [ χ ] ) S_{2}^{\mathrm{new}}(315, [\chi]) S 2 n e w ( 3 1 5 , [ χ ] ) .
p p p
F p ( T ) F_p(T) F p ( T )
2 2 2
T 4 − 2 T 3 + ⋯ + 1 T^{4} - 2 T^{3} + \cdots + 1 T 4 − 2 T 3 + ⋯ + 1
T^4 - 2*T^3 + 5*T^2 + 2*T + 1
3 3 3
T 4 T^{4} T 4
T^4
5 5 5
( T 2 − T + 1 ) 2 (T^{2} - T + 1)^{2} ( T 2 − T + 1 ) 2
(T^2 - T + 1)^2
7 7 7
T 4 − 2 T 3 + ⋯ + 49 T^{4} - 2 T^{3} + \cdots + 49 T 4 − 2 T 3 + ⋯ + 4 9
T^4 - 2*T^3 - 3*T^2 - 14*T + 49
11 11 1 1
T 4 − 4 T 3 + ⋯ + 16 T^{4} - 4 T^{3} + \cdots + 16 T 4 − 4 T 3 + ⋯ + 1 6
T^4 - 4*T^3 + 20*T^2 + 16*T + 16
13 13 1 3
( T 2 + 4 T − 4 ) 2 (T^{2} + 4 T - 4)^{2} ( T 2 + 4 T − 4 ) 2
(T^2 + 4*T - 4)^2
17 17 1 7
T 4 − 4 T 3 + ⋯ + 16 T^{4} - 4 T^{3} + \cdots + 16 T 4 − 4 T 3 + ⋯ + 1 6
T^4 - 4*T^3 + 20*T^2 + 16*T + 16
19 19 1 9
T 4 + 8 T 2 + 64 T^{4} + 8T^{2} + 64 T 4 + 8 T 2 + 6 4
T^4 + 8*T^2 + 64
23 23 2 3
T 4 + 2 T 3 + ⋯ + 1 T^{4} + 2 T^{3} + \cdots + 1 T 4 + 2 T 3 + ⋯ + 1
T^4 + 2*T^3 + 5*T^2 - 2*T + 1
29 29 2 9
( T − 1 ) 4 (T - 1)^{4} ( T − 1 ) 4
(T - 1)^4
31 31 3 1
( T 2 − 6 T + 36 ) 2 (T^{2} - 6 T + 36)^{2} ( T 2 − 6 T + 3 6 ) 2
(T^2 - 6*T + 36)^2
37 37 3 7
T 4 T^{4} T 4
T^4
41 41 4 1
( T 2 − 10 T + 17 ) 2 (T^{2} - 10 T + 17)^{2} ( T 2 − 1 0 T + 1 7 ) 2
(T^2 - 10*T + 17)^2
43 43 4 3
( T 2 − 10 T + 23 ) 2 (T^{2} - 10 T + 23)^{2} ( T 2 − 1 0 T + 2 3 ) 2
(T^2 - 10*T + 23)^2
47 47 4 7
( T 2 − 2 T + 4 ) 2 (T^{2} - 2 T + 4)^{2} ( T 2 − 2 T + 4 ) 2
(T^2 - 2*T + 4)^2
53 53 5 3
T 4 + 8 T 3 + ⋯ + 64 T^{4} + 8 T^{3} + \cdots + 64 T 4 + 8 T 3 + ⋯ + 6 4
T^4 + 8*T^3 + 56*T^2 + 64*T + 64
59 59 5 9
T 4 + 8 T 3 + ⋯ + 3136 T^{4} + 8 T^{3} + \cdots + 3136 T 4 + 8 T 3 + ⋯ + 3 1 3 6
T^4 + 8*T^3 + 120*T^2 - 448*T + 3136
61 61 6 1
T 4 − 6 T 3 + ⋯ + 3969 T^{4} - 6 T^{3} + \cdots + 3969 T 4 − 6 T 3 + ⋯ + 3 9 6 9
T^4 - 6*T^3 + 99*T^2 + 378*T + 3969
67 67 6 7
T 4 + 22 T 3 + ⋯ + 14161 T^{4} + 22 T^{3} + \cdots + 14161 T 4 + 2 2 T 3 + ⋯ + 1 4 1 6 1
T^4 + 22*T^3 + 365*T^2 + 2618*T + 14161
71 71 7 1
( T 2 − 8 T − 56 ) 2 (T^{2} - 8 T - 56)^{2} ( T 2 − 8 T − 5 6 ) 2
(T^2 - 8*T - 56)^2
73 73 7 3
T 4 + 4 T 3 + ⋯ + 16 T^{4} + 4 T^{3} + \cdots + 16 T 4 + 4 T 3 + ⋯ + 1 6
T^4 + 4*T^3 + 20*T^2 - 16*T + 16
79 79 7 9
T 4 + 24 T 3 + ⋯ + 18496 T^{4} + 24 T^{3} + \cdots + 18496 T 4 + 2 4 T 3 + ⋯ + 1 8 4 9 6
T^4 + 24*T^3 + 440*T^2 + 3264*T + 18496
83 83 8 3
( T 2 + 2 T − 161 ) 2 (T^{2} + 2 T - 161)^{2} ( T 2 + 2 T − 1 6 1 ) 2
(T^2 + 2*T - 161)^2
89 89 8 9
T 4 + 6 T 3 + ⋯ + 529 T^{4} + 6 T^{3} + \cdots + 529 T 4 + 6 T 3 + ⋯ + 5 2 9
T^4 + 6*T^3 + 59*T^2 - 138*T + 529
97 97 9 7
( T 2 − 12 T + 4 ) 2 (T^{2} - 12 T + 4)^{2} ( T 2 − 1 2 T + 4 ) 2
(T^2 - 12*T + 4)^2
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